ideas. You can see an edited video of
part of this lesson (from Illustrative
Mathematics, the Smarter Balanced
Assessment Consortium and Teaching
Channel) at www.teachingchannel.
org/videos/ratios-and-proportions-lesson-sbac.

Starting with a Mistake:Are the Coordinates Correct?

A pre-calculus teacher puts a graph
on the board with some coordinates
labeled in two different colors. The
teacher tells students there might be
an error in the coordinates shown in
red. Students work in pairs to discuss
the posted work, considering whether
there is a mistake and determining
how they will make their case to the
rest of the class. The teacher then
convenes the class for a large-group
discussion in which the students
present their thinking to their peers,
eventually coming to agreement about
the correct solution. (You can see
an edited video of this lesson from
PBS Learning Media at http://mass.
pbslearningmedia.org/resource/mtc13.
pd.math.deb/encouraging-debate.)

What We Can Learn from TheseUpside-Down Classrooms

Short edited video excerpts of classrooms like these may not show all the
elements of an upside-down lesson.

In some of the full lessons for which
excerpts are shown in the video clips
above, for example, we can assume
that the teacher helped students crystallize the mathematical conclusion
at the end of the lesson (off screen).

What we can notice across these
examples, however, are the types of
tasks the teachers have chosen and
the ways the teachers orchestrate the
classroom discourse.

In each of these classrooms, the
teacher sets the stage with the whole
group, elaborating the task or facilitating students in formulating the
question they will try to answer.

All four types of tasks used in theseexamples can readily be adapted toany grade level, and there are likelyother types of problems or tasks thatwould also work well for upside-down lessons. In choosing tasks forsuch lessons, teachers look for “low-floor high-ceiling” tasks. This meanslooking for tasks with multiple entrypoints—so that essentially all studentscan access the task at some level—butthat also allow for considerable depthor extension (Smith & Stein, 2011).In terms of orchestrating discourse,the teachers in these classrooms moveamong students as they work, askingquestions or offering comments like,“I notice that in your group you havethree different models. I’ll be back in afew minutes to see if you have agreedon which model you want to presentto the class,” or “Can you draw onyour paper a picture of what you justsaid?” or “How did you decide todivide by 7?” When the teacher bringsstudents together after their groupwork, students present their findingsand solutions to the whole class, withthe teacher asking clarifying ques-tions, facilitating further discussion,and, finally, making explicit the math-ematical connection between students’work and the mathematical goal ofthe lesson.

We also notice that sometimesstudents in these classrooms shareanswers or approaches that areincorrect. Teachers have learned thatvaluable classroom discussions canarise from wrong answers. Jo Boaler(2015) suggests that we actually learnmore from making a mistake thanfrom getting a right answer. Upside-down teaching helps both studentsand teachers understand that mistakeswill happen, and that when they do,the class will use the opportunity todig into the thinking that led to themistake, leading to deeper under-standing of the mathematics andincreasing the likelihood that studentswill be able to use what they’ve learnedto solve other problems in the future.Teachers today have access to agrowing body of publicly availableclassroom videos showing this kind ofmathematics teaching, whether labeledupside-down, problem-centered, student-focused, or just math class. Videos suchas those described here provide a greatopportunity for individual reflectionor professional discussion among col-leagues. In looking at classrooms inreal time or analyzing online videos,educators can ask questions like,n What kind of problem or task doesthe teacher use to start the lesson?n How does the teacher encouragestudents’ thinking and stimulatestudent discourse?n What kinds of questions does theteacher ask?n What do you notice about theroles of the teacher and student?n How does the teacher sequencestudents’ presentation of their work?n How does the teacher connect theclass discussion to the mathematicaloutcome of the lesson?n How is this classroom similar to ordifferent from your classroom or theother classrooms you see?Not all upside-down classrooms willcomplete a lesson in one class periodor follow the same format. Effectiveupside-down classrooms differnoticeably in terms of how they’reIn upside-down teaching, students learnabout the power of effort and persistence,become more confident problem solvers,and even grow their intelligence.